1 Lowen and Teich, JASA 1 Auditory-nerve action potentials form a non-renewal point process over short as well as long time scales Steven B. Lowen and Malvin C. Teich Department of Electrical Engineering 500 West 120th Street Columbia University, New York, NY Abstract The ring patterns of auditory-nerve action potentials exhibit long-term fractal uctuations that do not arise from the distribution of the interevent intervals, but rather from the ordering of these intervals. Using the serial interevent-interval correlation coecient, the Fano-factor time curve, and shuing of interevent intervals, we show that adjacent intervals for spontaneous rings exhibit signicant correlation. The events are therefore non-renewal over short as well as long time scales. PACS Numbers: Pg, Yp, Bt Introduction It is well known that in mammals, the pathway for the transfer of information from the inner ear to higher auditory centers in the brain is provided by the VIIIth nerve. The neural signal transmitted on individual bers of this nerve has been studied by many researchers, with the goal of gaining insight into the mechanisms of information encoding (Galambos and Davis, 1943, 1948; Tasaki, 1954; Katsuki et al., 1958; Kiang et al., 1962, 1965; Rose et al., 1967, 1971; Hind et al., 1967; Evans, 1972, 1975; Kiang, 1984, Teich and Khanna, 1985; Young and Barta, 1986; Teich, 1989, 1992). This signal comprises a series of brief electrical nerve spikes, whose amplitude and energy are widely assumed not to be signicant variables. Rather, it is generally accepted that the times of occurrences of the spikes carry the auditory information. Randomness is directly involved, since ensembles of identical single-ber experiments lead to diering sequences of nerve spikes (Tasaki, 1954; Peake et al., 1962; Rupert et al., 1963; Kiang, 1984). From a mathematical point of view, the neural activity in a peripheral auditory ber is perhaps best characterized as an (unmarked) stochastic point process (Parzen, 1962). To take the simplest case, we consider auditory neural rings in the absence of any external acoustic stimulation. Auditory neurons spontaneously re under such conditions, albeit at
2 Lowen and Teich, JASA 2 a lower rate than if a stimulus were present. Any realistic model of neural spike train data must include the eects of dead time (absolute refractoriness), which limits the rate at which neurons can re. After a relatively brief (1-2 ms) dead-time interval, the neuron is ready to re again, and the time to the next ring event approximates an exponentially distributed random variable. (More precisely, the neuron recovers gradually, over a relative refractory period that lasts tens of milliseconds, rather than abruptly.) Figure 1 provides a histogram of the interevent times for the spontaneous spike train from cat auditory nerve unit A (which is typical) on a semilogarithmic scale; the curve approximates an exponential distribution after the dead-time period. The dead-time modied Poisson point process (DTMP) (Ricciardi and Esposito, 1966; Prucnal and Teich, 1983; Teich, 1985), which is a renewal point process, also has exponentially distributed interevent times after the dead-time period, and has been used as a zeroth-order approximation to model such data. Two related renewal point process that provide somewhat better models make use of stochastic dead time (Teich, Matin, and Cantor, 1978; Young and Barta, 1986) and relative dead time, also called sick time (Gray, 1967; Teich and Diament, 1980; Gaumond et al., 1982). 1 Long-Term Correlations However, despite the exponential character of the interevent-time histogram, there now exists compelling evidence that, over long time scales, auditory-nerve spike-train data are not renewal (Teich, 1989; Teich, Turcott, and Lowen, 1990; Teich, Johnson, Kumar, and Turcott, 1990; Woo, 1990; Powers, 1991; Woo, Sachs, Teich, 1992; Teich 1992), but that long-term positive correlations are present in the data. One measure of this correlation is the Fanofactor time curve (FFC), dened as the variance of the n umber of events in a specied counting time T divided by the mean number of events in that counting time. In general the FFC varies with the counting time T. For a DTMP process, the Fano factor never exceeds unity for any counting time, but for the auditory-nerve spike-train data the Fano factor increases in a power-law fashion beyond unity for large counting times. Figure 2 shows the experimental FFC for unit A (solid curve). Note that the Fano factor increases steadily for counting times greater than about 100 milliseconds, and exceeds 10 for counting times in excess of 30 seconds. This steady increase indicates th e presence of a fractal process, with uctuations on many time scales, rather than a collection of a few simple processes with single time constants, such as breathing or heartbeat. A plot of the FFC alone does not determine whether this large Fano factor arises from the distribution of the interevent times or their order. This issue is resolved by shuing (randomly reordering) the interevent intervals and then replotting the Fano factor vs. counting time. FFCs constructed from shued data yield information about the relative sizes of the intervals only; all correlation and dependencies among the intervals are destroyed by shuing. The non-solid curves in Fig. 2 illustrate ve successive shuings. The shued interevent intervals have
3 Lowen and Teich, JASA 3 FFCs which all approach a value less than unity for large counting times, illustrating that it is the ordering of the intervals that gives rise to the growth in the FFC. All auditory nerve-ber data examined to date (for which long-duration spike trains exist) show this type of behavior, both in the presence and in the absence of stimulation. Thus the auditory nerve data is more clustered than the DTMP process. The interevent intervals will be relatively more similar within a long counting time T than in the DTMP process, leading to a larger variation among counting times, and a larger Fano factor. The FFC provides a sensitive test for the presence of clusters, detecting them even when they are interleaved and therefore not readily apparent in visual representations of the auditorynerve rings. Doubly stochastic models, such as the fractal-s hot-noise-driven Poisson point process (FSNDP) (Lowen and Teich, 1991) are not renewal, and t auditory nerve-ber data well. The FSNDP model contains interdigitated clusters, much as the auditory nerve-ber data appear to. For a variety of statistical measures, the data and simulations of the FSNDP yield nearly identical results, particularly for time scales of 100 milliseconds or larger (Teich, Turcott, and Lowen, 1990). 2 Short-Term Correlations Aside from the positive correlation over long times, there is also a small but signicant correlation over times scales on the order of tens of milliseconds. Figure 3, which is a magnication of Fig. 2 (but plotted with linear coordinates), shows this eect. Again, the shued intervals are repeatably dierent from the unshued intervals, and again they are closer to unity. For this particular unit, and in this range of counting times T, the FFC increases with shuing, indicating anti-clustered interevent intervals. Thus for this unit, long intervals are likely to be followed by short intervals and vice versa; over long time scales, as with all auditory units, the opposite is true. Many of the units studied exhibit this short-time eect. This behavior is demonstrated in Table 1 using another measure, the serial intereventinterval correlation coecient (SIICC). The SIICC is the correlation coecient calculated between adjacent intervals, and is dened by (N? 2)?1 X N?1 i=1 (N? 1)?1 ( i? h i)( i+1? h i) NX i=1 ( i? h i) 2
4 Lowen and Teich, JASA 4 where N is the number of intervals, h i N?1 is the sample average interevent time, and the i represent the individual interevent intervals. The SIICC ranges from -1 (perfect anticorrelation) to +1 (perfect correlation), and is zero for uncorrelated intervals. The SIICC is calculated for the data before and after shuing, and is presented in columns 4 and 5 of the table, respectively. Shuing, i. e. the random rearrangement of the interevent intervals, provides a useful check on the SIICC computation algorithm. A random collection of intervals should exhibit no signicant correlation. Similar results for the SIICC (for unshued data only) have been reported by Cooper (1989). These SIICC values do not, by themselves, prove the existence or absence of correlation, so we develop a test of signicance. Suppose that a number of independent interevent intervals are drawn from the same distribution, and that the SIICC is calculated for these intervals. Repeating this procedure will in general yield a dierent SIICC value, depending on the particular intervals selected. Thus the SIICC statistic has a margin of error. For the case of a large number of independent intervals, the SIICC will have a Gaussian distribution with zero mean and a variance equal to the inverse of the number of intervals. This was veried by estimating the SIICC for 10 4 sets of 10 4 simulated exponentially distributed interevent-time samples, 10 4 sets of 10 3 samples, and 10 4 sets of 10 2 samples, all with the same mean. Exponentially distributed random variables have a nite mean and variance, so that the Central Limit Theorem may be applied. For each SIICC value we compute the probability that a random collection of intervals would generate an SIICC of that magnitude or larger. The results are displayed in columns 6 and 7 of the table for unshued and shued intervals, respectively. A relatively large probability indicates that the corresponding data set is probably uncorrelated, while a small one indicates the existence of correlation. The results show that the ten shued data sets are not signicantly correlated, as expected, since shuing destroys any correlation; however, most of the unshued data sets do exhibit signicant correlation. Interestingly, there is no clear trend in the data. Some data sets exhibit signicant positive correlation, some signicant negative correlation, and some no signicant correlation at all. The correlation does not seem to be related to the average interevent interval. Thus no simple model is likely to suce in describing the relationship between nearby interevent intervals. The fractal behavior underlying the long-term correlations is almost certainly not related to the short-term correlations. The fractal part of the Fano factor, as an example, scales as T, with 0 < < 1, and therefore has only minimal importance over the relatively short time scale of a single interevent interval. The FSNDP model we employed to simulate the auditorynerve spike behavior of unit A yielded an SIICC that was only slightly statistically signicant. NX i=1 i
5 Lowen and Teich, JASA 5 For 100 runs, each using a dierent random-number seed, the computed likelihoods were all greater than 10?5. The FSNDP produced a slight positive correlation in the interval statistics as expected, much less signicant than that shown in Table 1 for unit A. Furthermore, the most signicant correlations shown in the table are negative, while the FSNDP and other similar fractal models necessarily introduce a positive correlation. Instead, perhaps the short-term correlation arises from a form of refractoriness with memory that extends over a few interevent times. Such models have been successfully employed for clicks (Gaumond et al., 1983), and tone bursts (Lutkenhoner and Smith, 1986), for which non-renewal, and indeed non-stationary, behavior would be expected. 3 Discussion Auditory-nerve ring patterns are not well modeled as a dead-time-modied Poisson process (DTMP), either in the absence or in the presence of stimulation. Over long time scales, the ring patterns exhibit long-term fractal uctuations that do not arise from the distribution of the interevent intervals, but rather from their ordering. Using the SIICC, shuing, and the FFC, we have shown that adjacent intervals have signicant correlation, and thus the process is also non-renewal at short times. We computed the SIICC both before and after shuing, eliminating the possibility that the correlation is an artifact of the interevent-time distribution. Distributions with long tails can generate apparent correlation statistics in nite data samples even when no such correlation exists. The non-renewal nature at short times is also apparent in higher-order interevent time distributions (Teich and Khanna, 1985). The Fano factor provides a dierent picture of the relationships between closely spaced interevent times, showing short-time correlations as well as long-term behavior simultaneously. The dierence between the shued and unshued FFCs provide further evidence for the nonrenewal nature of the auditory nerve-ber data over all time scales. Cognizance of this fact should lead to an improved understanding of the physiology underlying the generation of sequences of auditory-nerve action potentials. Acknowledgement This work was supported by the Oce of Naval Research under Grant No. N J-1251.
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9 Lowen and Teich, JASA 9 Table Average interval Number SIICC Likelihood Unit (ms) of events unshued shued unshued shued A :08 10?9 6:12 10?1 B :93 10?8 6:61 10?1 C :08 10?1 2:72 10?1 D :58 10?9 4:68 10?1 E :33 10?1 6:62 10?1 F :56 10?1 2:11 10?1 G :02 10?5 1:91 10?1 H :28 10?6 9:67 10?1 I :94 10?3 9:57 10?1 J :32 10?2 4:83 10?1 Figure Captions Figure 1: Semilogarithmic plot of relative frequency of interevent times (interevent-time histogram) for unit A. The data closely follow a straight line for interevent times greater than about 4 milliseconds. Bin width is 0.5 milliseconds. The CF for this unit is 10.2 khz. Figure 2: Doubly logarithmic plot of Fano-factor time curve (FFC) for unit A. The curve for the unshued data approximates a straight line of positive slope for counting times larger than several hundred milliseconds, whereas the curve for the shued data does not rise, indicating that the action potentials are clustered. Figure 3: Doubly linear plot of Fano factor time curve (FFC) for unit A for counting times less than 100 milliseconds. The curves for the shued data resemble each other but are consistently higher than the curve for the unshued data.